39ed4

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39ed4

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Prime factorization	3 × 13
Step size	61.5385 ¢
Octave	20\39ed4 (1230.77 ¢)
Twelfth	31\39ed4 (1907.69 ¢)
Consistency limit	1
Distinct consistency limit	1

39 equal divisions of the 4th harmonic (abbreviated 39ed4) is a nonoctave tuning system that divides the interval of 4/1 into 39 equal parts of about 61.5 ¢ each. Each step represents a frequency ratio of 4^1/39, or the 39th root of 4.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	61.5	27/26
2	123.1	29/27
3	184.6	10/9, 29/26
4	246.2	15/13
5	307.7
6	369.2	21/17, 26/21
7	430.8	9/7
8	492.3
9	553.8	29/21
10	615.4	10/7, 27/19
11	676.9
12	738.5	23/15, 26/17, 29/19
13	800	27/17
14	861.5
15	923.1	17/10, 29/17
16	984.6	23/13
17	1046.2
18	1107.7	17/9, 19/10
19	1169.2
20	1230.8
21	1292.3	19/9
22	1353.8
23	1415.4	25/11
24	1476.9
25	1538.5	17/7
26	1600
27	1661.5	13/5
28	1723.1	19/7, 27/10
29	1784.6
30	1846.2	29/10
31	1907.7	3/1
32	1969.2
33	2030.8	29/9
34	2092.3
35	2153.8
36	2215.4
37	2276.9	26/7
38	2338.5	27/7
39	2400

Harmonics

Approximation of harmonics in 39ed4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+30.8	+5.7	+0.0	-17.1	-25.0	+15.8	+30.8	+11.5	+13.7	-28.2	+5.7
	Relative (%)	+50.0	+9.3	+0.0	-27.8	-40.7	+25.7	+50.0	+18.6	+22.2	-45.9	+9.3
Steps (reduced)		20 (20)	31 (31)	39 (0)	45 (6)	50 (11)	55 (16)	59 (20)	62 (23)	65 (26)	67 (28)	70 (31)

Approximation of harmonics in 39ed4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-9.8	-15.0	-11.3	+0.0	+18.1	-19.3	+10.2	-17.1	+21.5	+2.5	-12.9
	Relative (%)	-15.9	-24.3	-18.4	+0.0	+29.4	-31.4	+16.5	-27.8	+35.0	+4.1	-20.9
Steps (reduced)		72 (33)	74 (35)	76 (37)	78 (0)	80 (2)	81 (3)	83 (5)	84 (6)	86 (8)	87 (9)	88 (10)